3.2259 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=383 \[ -\frac{5 c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (6 b e g-13 c d g+c e f)}{8 e^2 \sqrt{d+e x} (2 c d-b e)}+\frac{5 c^2 (6 b e g-13 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{8 e^2 \sqrt{2 c d-b e}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^{13/2} (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (6 b e g-13 c d g+c e f)}{12 e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (6 b e g-13 c d g+c e f)}{24 e^2 (d+e x)^{5/2} (2 c d-b e)} \]

[Out]

(-5*c^2*(c*e*f - 13*c*d*g + 6*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/
(8*e^2*(2*c*d - b*e)*Sqrt[d + e*x]) - (5*c*(c*e*f - 13*c*d*g + 6*b*e*g)*(d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(24*e^2*(2*c*d - b*e)*(d + e*x)^(5/2)) + ((
c*e*f - 13*c*d*g + 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(12*e^2
*(2*c*d - b*e)*(d + e*x)^(9/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*
x^2)^(7/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^(13/2)) + (5*c^2*(c*e*f - 13*c*d*g +
6*b*e*g)*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b*e]*Sq
rt[d + e*x])])/(8*e^2*Sqrt[2*c*d - b*e])

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Rubi [A]  time = 1.30279, antiderivative size = 383, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.109 \[ -\frac{5 c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (6 b e g-13 c d g+c e f)}{8 e^2 \sqrt{d+e x} (2 c d-b e)}+\frac{5 c^2 (6 b e g-13 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{8 e^2 \sqrt{2 c d-b e}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^{13/2} (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (6 b e g-13 c d g+c e f)}{12 e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (6 b e g-13 c d g+c e f)}{24 e^2 (d+e x)^{5/2} (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(13/2),x]

[Out]

(-5*c^2*(c*e*f - 13*c*d*g + 6*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/
(8*e^2*(2*c*d - b*e)*Sqrt[d + e*x]) - (5*c*(c*e*f - 13*c*d*g + 6*b*e*g)*(d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(24*e^2*(2*c*d - b*e)*(d + e*x)^(5/2)) + ((
c*e*f - 13*c*d*g + 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(12*e^2
*(2*c*d - b*e)*(d + e*x)^(9/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*
x^2)^(7/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^(13/2)) + (5*c^2*(c*e*f - 13*c*d*g +
6*b*e*g)*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b*e]*Sq
rt[d + e*x])])/(8*e^2*Sqrt[2*c*d - b*e])

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Rubi in Sympy [A]  time = 161.781, size = 357, normalized size = 0.93 \[ - \frac{5 c^{2} \left (6 b e g - 13 c d g + c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{8 e^{2} \sqrt{b e - 2 c d}} + \frac{5 c^{2} \left (6 b e g - 13 c d g + c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{8 e^{2} \sqrt{d + e x} \left (b e - 2 c d\right )} + \frac{5 c \left (6 b e g - 13 c d g + c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{24 e^{2} \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )} - \frac{\left (6 b e g - 13 c d g + c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{12 e^{2} \left (d + e x\right )^{\frac{9}{2}} \left (b e - 2 c d\right )} - \frac{\left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{3 e^{2} \left (d + e x\right )^{\frac{13}{2}} \left (b e - 2 c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(13/2),x)

[Out]

-5*c**2*(6*b*e*g - 13*c*d*g + c*e*f)*atan(sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e
+ c*d))/(sqrt(d + e*x)*sqrt(b*e - 2*c*d)))/(8*e**2*sqrt(b*e - 2*c*d)) + 5*c**2*
(6*b*e*g - 13*c*d*g + c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(8*e
**2*sqrt(d + e*x)*(b*e - 2*c*d)) + 5*c*(6*b*e*g - 13*c*d*g + c*e*f)*(-b*e**2*x -
c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(24*e**2*(d + e*x)**(5/2)*(b*e - 2*c*d)) -
(6*b*e*g - 13*c*d*g + c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/
(12*e**2*(d + e*x)**(9/2)*(b*e - 2*c*d)) - (d*g - e*f)*(-b*e**2*x - c*e**2*x**2
+ d*(-b*e + c*d))**(7/2)/(3*e**2*(d + e*x)**(13/2)*(b*e - 2*c*d))

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Mathematica [A]  time = 2.02357, size = 230, normalized size = 0.6 \[ \frac{((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{-\frac{3 c (18 b e g-47 c d g+11 c e f)}{d+e x}-\frac{2 (b e-2 c d) (6 b e g-25 c d g+13 c e f)}{(d+e x)^2}+\frac{8 (b e-2 c d)^2 (d g-e f)}{(d+e x)^3}+48 c^2 g}{(b e-c d+c e x)^2}+\frac{15 c^2 (6 b e g-13 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{\sqrt{2 c d-b e} (c (d-e x)-b e)^{5/2}}\right )}{24 e^2 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(13/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((48*c^2*g + (8*(-2*c*d + b*e)^2*(-(e*
f) + d*g))/(d + e*x)^3 - (2*(-2*c*d + b*e)*(13*c*e*f - 25*c*d*g + 6*b*e*g))/(d +
e*x)^2 - (3*c*(11*c*e*f - 47*c*d*g + 18*b*e*g))/(d + e*x))/(-(c*d) + b*e + c*e*
x)^2 + (15*c^2*(c*e*f - 13*c*d*g + 6*b*e*g)*ArcTanh[Sqrt[c*d - b*e - c*e*x]/Sqrt
[2*c*d - b*e]])/(Sqrt[2*c*d - b*e]*(-(b*e) + c*(d - e*x))^(5/2))))/(24*e^2*(d +
e*x)^(5/2))

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Maple [B]  time = 0.044, size = 1070, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(13/2),x)

[Out]

-1/24*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(8*b^2*e^3*f*(b*e-2*c*d)^(1/2)*(-c*
e*x-b*e+c*d)^(1/2)-121*c^2*d^3*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+15*arc
tan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*c^3*e^4*f+15*arctan((-c*e*x-b*
e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^3*e*f-195*arctan((-c*e*x-b*e+c*d)^(1/2)/(b
*e-2*c*d)^(1/2))*c^3*d^4*g-48*x^3*c^2*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(
1/2)+26*x*b*c*e^3*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-326*x*c^2*d^2*e*g*(
b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+14*x*c^2*d*e^2*f*(b*e-2*c*d)^(1/2)*(-c*e
*x-b*e+c*d)^(1/2)+12*b*c*d^2*e*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-6*b*c*
d*e^2*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+270*arctan((-c*e*x-b*e+c*d)^(1/
2)/(b*e-2*c*d)^(1/2))*x^2*b*c^2*d*e^3*g+270*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2
*c*d)^(1/2))*x*b*c^2*d^2*e^2*g+54*x^2*b*c*e^3*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*
d)^(1/2)-285*x^2*c^2*d*e^2*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+33*x^2*c^2
*e^3*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+12*x*b^2*e^3*g*(b*e-2*c*d)^(1/2)
*(-c*e*x-b*e+c*d)^(1/2)+4*b^2*d*e^2*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+1
3*c^2*d^2*e*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+90*arctan((-c*e*x-b*e+c*d
)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*b*c^2*e^4*g-195*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*
e-2*c*d)^(1/2))*x^3*c^3*d*e^3*g-585*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1
/2))*x^2*c^3*d^2*e^2*g+45*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c
^3*d*e^3*f-585*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^3*d^3*e*g+45
*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^3*d^2*e^2*f+90*arctan((-c*
e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^3*e*g+34*x*b*c*d*e^2*g*(b*e-2*c*d)
^(1/2)*(-c*e*x-b*e+c*d)^(1/2))/(e*x+d)^(7/2)/(-c*e*x-b*e+c*d)^(1/2)/e^2/(b*e-2*c
*d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.41603, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(13/2),x, algorithm="fricas")

[Out]

[1/48*(15*(c^3*d^2*e*f + (c^3*e^3*f - (13*c^3*d*e^2 - 6*b*c^2*e^3)*g)*x^2 - (13*
c^3*d^3 - 6*b*c^2*d^2*e)*g + 2*(c^3*d*e^2*f - (13*c^3*d^2*e - 6*b*c^2*d*e^2)*g)*
x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)*log((2*sqrt(-c*e^2*x
^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*d - b*e)*sqrt(e*x + d) - (c*e^2*x^2 - 3*c*d^2
+ 2*b*d*e - 2*(c*d*e - b*e^2)*x)*sqrt(2*c*d - b*e))/(e^2*x^2 + 2*d*e*x + d^2))
- 2*(48*c^3*e^4*g*x^4 - 3*(11*c^3*e^4*f - (79*c^3*d*e^3 - 2*b*c^2*e^4)*g)*x^3 +
((19*c^3*d*e^3 - 59*b*c^2*e^4)*f + (41*c^3*d^2*e^2 + 305*b*c^2*d*e^3 - 66*b^2*c*
e^4)*g)*x^2 + (13*c^3*d^3*e - 19*b*c^2*d^2*e^2 + 14*b^2*c*d*e^3 - 8*b^3*e^4)*f -
(121*c^3*d^4 - 133*b*c^2*d^3*e + 8*b^2*c*d^2*e^2 + 4*b^3*d*e^3)*g + ((c^3*d^2*e
^2 + 18*b*c^2*d*e^3 - 34*b^2*c*e^4)*f - (205*c^3*d^3*e - 348*b*c^2*d^2*e^2 + 26*
b^2*c*d*e^3 + 12*b^3*e^4)*g)*x)*sqrt(2*c*d - b*e))/((e^4*x^2 + 2*d*e^3*x + d^2*e
^2)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*sqrt(e*x + d)),
-1/24*(15*(c^3*d^2*e*f + (c^3*e^3*f - (13*c^3*d*e^2 - 6*b*c^2*e^3)*g)*x^2 - (13
*c^3*d^3 - 6*b*c^2*d^2*e)*g + 2*(c^3*d*e^2*f - (13*c^3*d^2*e - 6*b*c^2*d*e^2)*g)
*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)*arctan(sqrt(-c*e^2*
x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d + b*e)*sqrt(e*x + d)/(c*e^2*x^2 + b*e
^2*x - c*d^2 + b*d*e)) + (48*c^3*e^4*g*x^4 - 3*(11*c^3*e^4*f - (79*c^3*d*e^3 - 2
*b*c^2*e^4)*g)*x^3 + ((19*c^3*d*e^3 - 59*b*c^2*e^4)*f + (41*c^3*d^2*e^2 + 305*b*
c^2*d*e^3 - 66*b^2*c*e^4)*g)*x^2 + (13*c^3*d^3*e - 19*b*c^2*d^2*e^2 + 14*b^2*c*d
*e^3 - 8*b^3*e^4)*f - (121*c^3*d^4 - 133*b*c^2*d^3*e + 8*b^2*c*d^2*e^2 + 4*b^3*d
*e^3)*g + ((c^3*d^2*e^2 + 18*b*c^2*d*e^3 - 34*b^2*c*e^4)*f - (205*c^3*d^3*e - 34
8*b*c^2*d^2*e^2 + 26*b^2*c*d*e^3 + 12*b^3*e^4)*g)*x)*sqrt(-2*c*d + b*e))/((e^4*x
^2 + 2*d*e^3*x + d^2*e^2)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d
+ b*e)*sqrt(e*x + d))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(13/2),x, algorithm="giac")

[Out]

Timed out